Chapter 4 Motion in Two and Three Dimensions

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Chapter 4Motion in Two and Three Dimensions

• Position vector in three dimensions

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Instantaneous velocity
The direction of the instantaneous velocity
of a particle is tangent to the path at the
particles position.
The components of are
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Instantaneous acceleration

• The components of are

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4-5 Projectile Motion

• In projectile motion, the horizontal motion and
  the vertical motion are independent of each
  other that is, neither motion affects the other.
  A projectile with an initial velocity can
  be written as (see figure 4-10) -

• The horizontal motion has zero acceleration, and
  the vertical motion has a constant downward
  acceleration of - g.

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• The range R is the horizontal distance the
  projectile has traveled when it returns to its
  launch height.

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Examples of projectile motion
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Horizontal motion

• No acceleration

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Vertical motion (Equations of Motion )-
1 )
2 )
3 )
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The equation of the path

• In this equation, x0 0 and y0 0. The path, or
  trajectory, is a parabola. The angle is
  between and the direction.

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The horizontal range
To find t time of flight, y - y0 0 means
that

• This equation for R is only good if the final
  height equals the launch height. We have used the
  relationsin 2 2 sin cos .
• The range is a maximum when 45o

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Sample Problem 4-6

• In Fig. 4-15, a rescue plane flies at 198 km/h (
  55.0 m/s) and a constant elevation of 500 m
  toward a point directly over a boating accident
  victim struggling in the water. The pilot wants
  to release a rescue capsule so that it hits the
  water very close to the victim.

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(a)  What should be the angle of the pilot's
line of sight to the victim when the release is
made?
Solution

• Solving for t, we find t 10.1 s (take the
  positive root).

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(b)  As the capsule reaches the water, what is
its velocity in unit-vector notation and as
a magnitude and an angle?
When the capsule reaches the water,
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Sample Problem 4-7Figure 4-16 shows a pirate
ship 560 m from a fort defending the harbor
entrance of an island. A defense cannon, located
at sea level, fires balls at initial speed v0
82 m/s.

• (a)  At what angle from the horizontal
  must a ball be fired to hit the ship?

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SOLUTION 
Which gives
There are two solutions
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(b)  How far should the pirate ship be from the
cannon if it is to be beyond the maximum range of
the cannonballs?

• SOLUTION  Maximum range is -

The maximum range is 690m. Beyond that distance,
the ship is safe from the cannon.
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Sample Problem 4-8Figure 4-17 illustrates the
flight of Emanuel Zacchini over three Ferris
wheels, located as shown and each 18 m high.
Zacchini is launched with speed v0 26.5 m/s, at
an angle 53 up from the horizontal and
with an initial height of 3.0 m above the ground.
The net in which he is to land is at the same
height.

• (a)  Does he clear the first Ferris wheel?

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SOLUTION

• The equation of trajectory when x0 0 and y0 0
  is given by

Solving for y when x 23m gives
Since he begins 3m off the ground, he clears the
Ferris wheel by (23.3 18) 5.3 m
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(b)  If he reaches his maximum height when he is
over the middle Ferris wheel, what is his
clearance above it?

• SOLUTION 
• At maximum height, vy is 0. Therefore,

and he clears the middle Ferris wheel by (22.9
3.0 -18) m 7.9 m
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(c)  How far from the cannon should the center of
the net be positioned?

• SOLUTION 

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Avoid rounding errors

• One way to avoid rounding errors and other
  numerical errors is to solve problems
  algebraically, substituting numbers only in the
  final step.

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Uniform Circular Motion

• A particle is in uniform circular motion if it
  travels around a circle at uniform speed.
  Although the speed is uniform, the particle is
  accelerating.

• The acceleration is called a centripetal (center
  seeking) acceleration.

• T is called the period of revolution.

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• Thus, , which means that is directed
  along the radius r, pointing towards the circles
  center.

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Sample Problem 4-9

• Top gun pilots have long worried about taking a
  turn too tightly. As a pilot's body undergoes
  centripetal acceleration, with the head toward
  the center of curvature, the blood pressure in
  the brain decreases, leading to loss of brain
  function.
• There are several warning signs to signal a pilot
  to ease up when the centripetal acceleration is
  2g or 3g, the pilot feels heavy. At about 4g, the
  pilot's vision switches to black and white and
  narrows to tunnel vision. If that acceleration
  is sustained or increased, vision ceases and,
  soon after, the pilot is unconsciousa condition
  known as g-LOC for g-induced loss of
  consciousness.

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What is the centripetal acceleration, in g units,
of a pilot flying an F-22 at speed v 2500 km/h
(694 m/s) through a circular arc with radius of
curvature r 5.80 km?

• SOLUTION 

• If a pilot caught in a dogfight puts the aircraft
  into such a tight turn, the pilot goes into g-LOC
  almost immediately, with no warning signs to
  signal the danger.

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Relative Motion in One Dimension

• The coordinate xPA of P as measured by A is equal
  to the coordinate xPB of P as measured by B plus
  the coordinate xBA of B as measured by A. Note
  that x is a vector in one dimension.

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• The velocity vPA of P as measured by A is equal
  to the velocity vPB of P as measured by B plus
  the velocity vBA of B as measured by A. Note that
  v is a one dimensional vector. We have deleted
  the arrow on top.

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Because VBA is constant, the last term is zero.

• Observers on different frames of reference (that
  move at constant velocity relative to each other)
  will measure the same acceleration for a moving
  particle. Note that the acceleration is a one
  dimensional vector.

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Sample Problem 4-10For the situation of Fig.
4-20, Barbara's velocity relative to Alex is a
constant vBA 52 km/h and car P is moving in the
negative direction of the x axis.

• (a)  If Alex measures a constant velocity vPA
  -78 km/h for car P, what velocity vPB will
  Barbara measure?
• SOLUTION 

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(b)  If car P brakes to a stop relative to Alex
(and thus the ground) in time t 10 s at
constant acceleration, what is its acceleration
aPA relative to Alex?
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(c)  What is the acceleration aPB of car P
relative to Barbara during the braking?

• SOLUTION 
• To calculate the acceleration of car P relative
  to Barbara, we must use the car's velocities
  relative to Barbara. The initial velocity of P
  relative to Barbara is vPB -130 km/h. The final
  velocity of P relative to Barbara is -52 km/h
  (this is the velocity of the stopped car relative
  to the moving Barbara).

• This result is reasonable because Alex and
  Barbara have a constant relative velocity, they
  must measure the same acceleration.

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Relative Motion in Two Dimension

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Homework (due Sept 27)

• Question 10
• 17E
• 19E
• 25P
• 43E
• 51P